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The helicity of a particle is right-handed if the direction of its spin is the same as the direction of its motion. It is left-handed if the directions of spin and motion are opposite. So a standard clock, with its spin vector defined by the rotation of its hands, tossed with its face directed forwards, has left-handed helicity.

Mathematically, helicity is the sign of the projection of the spinvector onto the momentumvector: “left” is negative, “right” is positive.

The chirality of a particle is more abstract: It is determined by whether the particle transforms in a right- or left-handed representation of the Poincaré group.[a]

For massless particles – photons, gluons, and (hypothetical) gravitons – chirality is the same as helicity; a given massless particle appears to spin in the same direction along its axis of motion regardless of point of view of the observer.

For massive particles – such as electrons, quarks, and neutrinos – chirality and helicity must be distinguished: In the case of these particles, it is possible for an observer to change to a reference frame moving faster than the spinning particle, in which case the particle will then appear to move backwards, and its helicity (which may be thought of as “apparent chirality”) will be reversed.

A massless particle moves with the speed of light, so no real observer (who must always travel at less than the speed of light) can be in any reference frame where the particle appears to reverse its relative direction of spin, meaning that all real observers see the same helicity. Because of this, the direction of spin of massless particles is not affected by a change of viewpoint (Lorentz boost) in the direction of motion of the particle, and the sign of the projection (helicity) is fixed for all reference frames: The helicity of massless particles is a relativistic invariant (a quantity whose value is the same in all inertial reference frames) which always matches the massless particles' chirality.

The discovery of neutrino oscillation implies that neutrinos have mass, so the only observed massless particle is the photon. The gluon is also expected to be massless, although the assumption that it is has not been conclusively tested. Hence, these are the only two particles now known for which helicity could be identical to chirality, and only one of them has been confirmed by measurement. All other observed particles have mass and thus may have different helicities in different reference frames. It is still possible that as-yet unobserved particles, like the graviton, might be massless, and hence have invariant helicity that matches their chirality, like the photon.

Only left-handed fermions and right-handed antifermions interact with the weak interaction.[1]
In most circumstances, two left-handed fermions interact more strongly than right-handed or opposite-handed fermions, implying that the universe has a preference for left-handed chirality, which violates a symmetry of the other forces of nature.

Chirality for a Dirac fermionψ is defined through the operator γ5, which has eigenvalues ±1. Any Dirac field can thus be projected into its left- or right-handed component by acting with the projection operators½(1 − γ5) or ½(1 + γ5) on ψ.

The coupling of the charged weak interaction to fermions is proportional to the first projection operator, which is responsible for this interaction's parity symmetry violation.

A common source of confusion is due to conflating this operator with the helicity operator. Since the helicity of massive particles is frame-dependent, it might seem that the same particle would interact with the weak force according to one frame of reference, but not another. The resolution to this false paradox is that the chirality operator is equivalent to helicity for massless fields only, for which helicity is not frame-dependent. By contrast, for massive particles, chirality is not the same as helicity, so there is no frame dependence of the weak interaction: a particle that couples to the weak force in one frame does so in every frame.

A theory that is asymmetric with respect to chiralities is called a chiral theory, while a non-chiral (i.e., parity-symmetric) theory is sometimes called a vector theory. Many pieces of the Standard Model of physics are non-chiral, which is traceable to anomaly cancellation in chiral theories. Quantum chromodynamics is an example of a vector theory, since both chiralities of all quarks appear in the theory, and couple to gluons in the same way.

The exact nature of the neutrino is still unsettled and so the electroweak theories that have been proposed are somewhat different, but most accommodate the chirality of neutrinos in the same way as was already done for all other fermions.

Vector gauge theories with massless Dirac fermion fields ψ exhibit chiral symmetry, i.e., rotating the left-handed and the right-handed components independently makes no difference to the theory. We can write this as the action of rotation on the fields:

The chiral symmetry transformation can be divided into a component that treats the left-handed and the right-handed parts equally, known as vector symmetry, and a component that actually treats them differently, known as axial symmetry.[2] (cf. Current algebra.) A scalar field model encoding chiral symmetry and its breaking is the chiral model.

The most common application is expressed as equal treatment of clockwise and counter-clockwise rotations from a fixed frame of reference.

and it does not correspond to a conserved quantity, because it is explicitly violated by a quantum anomaly.

The remaining chiral symmetry SU(2)L×SU(2)R turns out to be spontaneously broken by a quark condensate⟨q¯RaqLb⟩=vδab{\displaystyle \textstyle \langle {\bar {q}}_{R}^{a}q_{L}^{b}\rangle =v\delta ^{ab}} formed through nonperturbative action of QCD gluons,
into the diagonal vector subgroup SU(2)V known as isospin. The Goldstone bosons corresponding to the three broken generators are the three pions.
As a consequence, the effective theory of QCD bound states like the baryons, must now include mass terms for them, ostensibly disallowed by unbroken chiral symmetry. Thus, this chiral symmetry breaking induces the bulk of hadron masses, such as those for the nucleons — in effect, the bulk of the mass of all visible matter.

In the real world, because of the nonvanishing and differing masses of the quarks, SU(2)L×SU(2)R is only an approximate symmetry[3] to begin with, and therefore the pions are not massless, but have small masses: they are pseudo-Goldstone bosons.[4]

Most usually, N = 3 is taken, the u, d, and s quarks taken to be light (the Eightfold way (physics)), so then approximately massless for the symmetry to be meaningful to a lowest order, while the other three quarks are sufficiently heavy to barely have a residual chiral symmetry be visible for practical purposes.

Here, SU(2)L (pronounced “SU(2) left”) is none other than SU(2)W from above, while B−L is the baryon number minus the lepton number. The electric charge formula in this model is given by

Q=I3L+I3R+B−L2{\displaystyle Q=I_{3L}+I_{3R}+{\frac {B-L}{2}}\,};

where I3L{\displaystyle \,I_{3L}\,} and I3R{\displaystyle \,I_{3R}\,} are the left and right weak isospin values of the fields in the theory.

There is also the chromodynamic SU(3)C. The idea was to restore parity by introducing a left-right symmetry. This is a group extension of Z2{\displaystyle \mathbb {Z} _{2}} (the left-right symmetry) by

The Higgs bosons needed to implement the breaking of left-right symmetry down to the Standard Model are

(1,3,1)2⊕(1,1,3)2{\displaystyle (1,3,1)_{2}\oplus (1,1,3)_{2}\,}.

This then provides three sterile neutrinos which are perfectly consistent with current[update]neutrino oscillation data. Within the seesaw mechanism, the sterile neutrinos become superheavy without affecting physics at low energies.

Because the left-right symmetry is spontaneously broken, left-right models predict domain walls. This left-right symmetry idea first appeared in the Pati–Salam model (1974), Mohapatra–Pati models (1975).

^Note, however, that representations such as Dirac spinors and others, have both right- and left-handed components. In cases like this, we can define projection operators that project out either the right or left hand components and discuss the right- and left-handed portions of the representation.

^Unlike the W+ and W− bosons, the neutral electroweak Z0 boson couples to both left and right-handed fermions, although not equally.

To see a summary of the differences and similarities between chirality and helicity (those covered here and more) in chart form, one may go to Pedagogic Aids to Quantum Field Theory and click on the link near the bottom of the page entitled "Chirality and Helicity Summary". To see an in depth discussion of the two with examples, which also shows how chirality and helicity approach the same thing as speed approaches that of light, click the link entitled "Chirality and Helicity in Depth" on the same page.